Definition 2.2 Let $\mathbin{\mathbb{C}}^{m}$ have complex coordinates $(z_{1},\dots,z_{m})$, and define a metric $g$, a real 2-form $\omega$ and a complex $m$-form $\Omega$ on $\mathbin{\mathbb{C}}^{m}$ by

$$\begin{split}g=|{\rm d}z_{1}|^{2}+\cdots+|{\rm d}z_{m}|^{2},\quad\omega&=\frac{i}{2}({\rm d}z_{1}\wedge{\rm d}\bar{z}_{1}+\cdots+{\rm d}z_{m}\wedge{\rm d}\bar{z}_{m}),\\ \text{and}\quad\Omega&={\rm d}z_{1}\wedge\cdots\wedge{\rm d}z_{m}.\end{split}$$

(1)

Then $\mathop{\rm Re}\Omega$ and $\mathop{\rm Im}\Omega$ are real $m$-forms on $\mathbin{\mathbb{C}}^{m}$. Let $L$ be an oriented real submanifold of $\mathbin{\mathbb{C}}^{m}$ of real dimension $m$. We say that $L$ is a special Lagrangian submanifold of $\mathbin{\mathbb{C}}^{m},$ or SL $m$-fold for short, if $L$ is calibrated with respect to $\mathop{\rm Re}\Omega$, in the sense of Definition 2.1.